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Fields

This quiz will test your knowledge of fields, which are algebraic structures that generalize the concept of number systems.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is a field?

  1. The set of integers
  2. The set of rational numbers
  3. The set of real numbers
  4. The set of complex numbers
Question 2 Multiple Choice (Single Answer)

What is the multiplicative identity of a field?

  1. 0
  2. 1
  3. -1
  4. 2
Question 3 Multiple Choice (Single Answer)

What is the additive inverse of an element in a field?

  1. The element itself
  2. The opposite of the element
  3. The reciprocal of the element
  4. The square root of the element
Question 4 Multiple Choice (Single Answer)

What is the characteristic of a field?

  1. The number of elements in the field
  2. The smallest positive integer such that $1 + 1 + ... + 1 = 0$
  3. The largest prime factor of the order of the field
  4. The number of generators of the field
Question 5 Multiple Choice (Single Answer)

Which of the following is not a field?

  1. The set of rational numbers
  2. The set of real numbers
  3. The set of complex numbers
  4. The set of quaternions
Question 6 Multiple Choice (Single Answer)

What is the order of a field?

  1. The number of elements in the field
  2. The smallest positive integer such that $1 + 1 + ... + 1 = 0$
  3. The largest prime factor of the order of the field
  4. The number of generators of the field
Question 7 Multiple Choice (Single Answer)

What is a generator of a field?

  1. An element of the field that generates the entire field under addition
  2. An element of the field that generates the entire field under multiplication
  3. An element of the field that generates the entire field under both addition and multiplication
  4. An element of the field that generates the entire field under neither addition nor multiplication
Question 8 Multiple Choice (Single Answer)

Which of the following is a finite field?

  1. The set of rational numbers
  2. The set of real numbers
  3. The set of complex numbers
  4. The set of integers modulo 7
Question 9 Multiple Choice (Single Answer)

What is the Galois group of a field?

  1. The group of automorphisms of the field
  2. The group of units of the field
  3. The group of generators of the field
  4. The group of elements of the field that have a multiplicative inverse
Question 10 Multiple Choice (Single Answer)

What is the fundamental theorem of Galois theory?

  1. Every finite field is a Galois field
  2. Every Galois field is a finite field
  3. The Galois group of a field is isomorphic to the group of permutations of the roots of a polynomial
  4. The Galois group of a field is isomorphic to the group of automorphisms of the field
Question 11 Multiple Choice (Single Answer)

Which of the following is a Galois field?

  1. The set of rational numbers
  2. The set of real numbers
  3. The set of complex numbers
  4. The set of integers modulo 7
Question 12 Multiple Choice (Single Answer)

What is a primitive element of a finite field?

  1. An element of the field that generates the entire field under addition
  2. An element of the field that generates the entire field under multiplication
  3. An element of the field that generates the entire field under both addition and multiplication
  4. An element of the field that generates the entire field under neither addition nor multiplication
Question 13 Multiple Choice (Single Answer)

Which of the following is not a property of a field?

  1. The field contains a multiplicative identity
  2. The field contains an additive inverse for every element
  3. The field is closed under addition and multiplication
  4. The field is ordered
Question 14 Multiple Choice (Single Answer)

What is the Wedderburn-Artin theorem?

  1. Every finite division ring is a field
  2. Every finite field is a division ring
  3. Every division ring is a field
  4. Every field is a division ring
Question 15 Multiple Choice (Single Answer)

Which of the following is not a field axiom?

  1. The field contains a multiplicative identity
  2. The field contains an additive inverse for every element
  3. The field is closed under addition and multiplication
  4. The field is associative under addition
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